We use the method of Lie groups, one of the powerful tools available to solve nonlinear PDEs, and which was discovered and applied
firstly by S. Lie in the nineteenth century, but only in the last decades has it become a common tool for both mathematicians
and physicists (see for examples [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}). The method consists of looking for the infinitesimal generators of a group of point transformations which
leave the equation under study invariant. An important point of the Lie theory is that the conditions for an equation to
admit a group of transformations are represented by a set of linear equations, the so-called “determining equations",
which are usually completely solvable. Having once found the groups of transformations, one can obtain a number of
interesting results, which include the possibility to reduce a partial differential equation with two independent variables to
an ordinary differential equation with one independent variable, etc.. Solving these reduced equations, one can obtain some particular
solutions for the original equations. These particular solutions are usually called “similarity solutions" or “invariant solutions" [1]}, [2]}, [3]}, [4]}. When the equation contains “arbitrary elements"(a variable coefficient deriving from the particular equation of state chosen to characterize the
physical mechanism. “Arbitrary elements" are functions or variable parameters
whose form is not strictly fixed and can be assigned
freely on the grounds of physical hypotheses about the nature
of the medium under consideration.), the theory gives rise to the problem of group classification of differential equations which is the core stone of modern group analysis [2]}, [4]}. In particular, in the past several years, a numbers of novel techniques, such as algebraic methods based on subgroup analysis of the equivalence group [17]}, [18]}, [19]}, [20]}, compatibility and direct
integration [2]}, [22]} (also referred as the Lie-Ovsiannikov method) as well as their generalizations
(eg. method of furcate split [23]}, additional and conditional
equivalence transformations [24]}, [25]}, extended and
generalized equivalence transformation group, gauging of arbitrary
elements by equivalence transformations [26]}, [27]}) have been proposed to solve
group classification problem for numerous nonlinear partial differential equations. Although a great deal of classification was solved by these methods, almost all of them are limited to the equations whose order are lower than four (see [27]} for details).
